Abstract

We analyze the form birefringence dispersion of periodic dielectric layered media. Using Bloch wave analysis, we investigate the shape of the normal surface and its dependence on wavelength. Our higher-order uniaxial approximation gives an explicit expression for the additional dispersion of the form birefringence that is due to the finite layer thickness.

© 1996 Optical Society of America

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References

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  1. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988); A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984); and references therein.
  2. L. M. Brekhovskikh, Waves in Layered Media (Academic, New York, 1960); M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980); and references therein.
  3. P. Yeh, W. J. Gunning, J. P. Eblen, M. Khoshnevisan, “Compensator for liquid crystal displays having two types of layers with different refractive indices alternating,” U.S. patent5,196,953 (1994).
  4. J. P. Eblen, W. J. Gunning, J. Beedy, D. Taber, L. Hale, P. Yeh, M. Khoshnevisan, SID Digest 94, 245 (1994).
  5. J. P. Eblen, W. J. Gunning, D. Taber, P. Yeh, M. Khoshnevisan, J. Beedy, L. Hale, Proc. SPIE 2262, 234 (1994).
    [CrossRef]
  6. C. Gu, P. Yeh, J. Opt. Soc. Am. B 12, 1094 (1995).
    [CrossRef]
  7. C. Gu, P. Yeh, in Digest of Annual Meeting (Optical Society of America, Washington, D.C., 1995), paper ThHH2.
  8. G. Campbell, R. K. Kostuk, J. Opt. Soc. Am A 12, 1113 (1995).
    [CrossRef]
  9. M. Bass ed., Handbook of Optics, 2nd ed. (McGraw-Hill, New York, 1994), Vol. 2, pp. 33.66–33.69.
  10. S.-T. Wu, “Nematic liquid crystals for active optics,” in Optical Materials— A Series of Advances, S. Musikant, B. J. Thompson, eds. (Dekker, New York, 1990), Vol. 1; S.-T. Wu, C.-S. Wu, W. Warenghem, M. Ismaili, Proc. SPIE 1815, 179 (1992).
    [CrossRef]

1995 (2)

G. Campbell, R. K. Kostuk, J. Opt. Soc. Am A 12, 1113 (1995).
[CrossRef]

C. Gu, P. Yeh, J. Opt. Soc. Am. B 12, 1094 (1995).
[CrossRef]

1994 (2)

J. P. Eblen, W. J. Gunning, J. Beedy, D. Taber, L. Hale, P. Yeh, M. Khoshnevisan, SID Digest 94, 245 (1994).

J. P. Eblen, W. J. Gunning, D. Taber, P. Yeh, M. Khoshnevisan, J. Beedy, L. Hale, Proc. SPIE 2262, 234 (1994).
[CrossRef]

Beedy, J.

J. P. Eblen, W. J. Gunning, J. Beedy, D. Taber, L. Hale, P. Yeh, M. Khoshnevisan, SID Digest 94, 245 (1994).

J. P. Eblen, W. J. Gunning, D. Taber, P. Yeh, M. Khoshnevisan, J. Beedy, L. Hale, Proc. SPIE 2262, 234 (1994).
[CrossRef]

Brekhovskikh, L. M.

L. M. Brekhovskikh, Waves in Layered Media (Academic, New York, 1960); M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980); and references therein.

Campbell, G.

G. Campbell, R. K. Kostuk, J. Opt. Soc. Am A 12, 1113 (1995).
[CrossRef]

Eblen, J. P.

J. P. Eblen, W. J. Gunning, J. Beedy, D. Taber, L. Hale, P. Yeh, M. Khoshnevisan, SID Digest 94, 245 (1994).

J. P. Eblen, W. J. Gunning, D. Taber, P. Yeh, M. Khoshnevisan, J. Beedy, L. Hale, Proc. SPIE 2262, 234 (1994).
[CrossRef]

P. Yeh, W. J. Gunning, J. P. Eblen, M. Khoshnevisan, “Compensator for liquid crystal displays having two types of layers with different refractive indices alternating,” U.S. patent5,196,953 (1994).

Gu, C.

C. Gu, P. Yeh, J. Opt. Soc. Am. B 12, 1094 (1995).
[CrossRef]

C. Gu, P. Yeh, in Digest of Annual Meeting (Optical Society of America, Washington, D.C., 1995), paper ThHH2.

Gunning, W. J.

J. P. Eblen, W. J. Gunning, D. Taber, P. Yeh, M. Khoshnevisan, J. Beedy, L. Hale, Proc. SPIE 2262, 234 (1994).
[CrossRef]

J. P. Eblen, W. J. Gunning, J. Beedy, D. Taber, L. Hale, P. Yeh, M. Khoshnevisan, SID Digest 94, 245 (1994).

P. Yeh, W. J. Gunning, J. P. Eblen, M. Khoshnevisan, “Compensator for liquid crystal displays having two types of layers with different refractive indices alternating,” U.S. patent5,196,953 (1994).

Hale, L.

J. P. Eblen, W. J. Gunning, D. Taber, P. Yeh, M. Khoshnevisan, J. Beedy, L. Hale, Proc. SPIE 2262, 234 (1994).
[CrossRef]

J. P. Eblen, W. J. Gunning, J. Beedy, D. Taber, L. Hale, P. Yeh, M. Khoshnevisan, SID Digest 94, 245 (1994).

Khoshnevisan, M.

J. P. Eblen, W. J. Gunning, J. Beedy, D. Taber, L. Hale, P. Yeh, M. Khoshnevisan, SID Digest 94, 245 (1994).

J. P. Eblen, W. J. Gunning, D. Taber, P. Yeh, M. Khoshnevisan, J. Beedy, L. Hale, Proc. SPIE 2262, 234 (1994).
[CrossRef]

P. Yeh, W. J. Gunning, J. P. Eblen, M. Khoshnevisan, “Compensator for liquid crystal displays having two types of layers with different refractive indices alternating,” U.S. patent5,196,953 (1994).

Kostuk, R. K.

G. Campbell, R. K. Kostuk, J. Opt. Soc. Am A 12, 1113 (1995).
[CrossRef]

Taber, D.

J. P. Eblen, W. J. Gunning, J. Beedy, D. Taber, L. Hale, P. Yeh, M. Khoshnevisan, SID Digest 94, 245 (1994).

J. P. Eblen, W. J. Gunning, D. Taber, P. Yeh, M. Khoshnevisan, J. Beedy, L. Hale, Proc. SPIE 2262, 234 (1994).
[CrossRef]

Wu, S.-T.

S.-T. Wu, “Nematic liquid crystals for active optics,” in Optical Materials— A Series of Advances, S. Musikant, B. J. Thompson, eds. (Dekker, New York, 1990), Vol. 1; S.-T. Wu, C.-S. Wu, W. Warenghem, M. Ismaili, Proc. SPIE 1815, 179 (1992).
[CrossRef]

Yeh, P.

C. Gu, P. Yeh, J. Opt. Soc. Am. B 12, 1094 (1995).
[CrossRef]

J. P. Eblen, W. J. Gunning, D. Taber, P. Yeh, M. Khoshnevisan, J. Beedy, L. Hale, Proc. SPIE 2262, 234 (1994).
[CrossRef]

J. P. Eblen, W. J. Gunning, J. Beedy, D. Taber, L. Hale, P. Yeh, M. Khoshnevisan, SID Digest 94, 245 (1994).

C. Gu, P. Yeh, in Digest of Annual Meeting (Optical Society of America, Washington, D.C., 1995), paper ThHH2.

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988); A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984); and references therein.

P. Yeh, W. J. Gunning, J. P. Eblen, M. Khoshnevisan, “Compensator for liquid crystal displays having two types of layers with different refractive indices alternating,” U.S. patent5,196,953 (1994).

J. Opt. Soc. Am A (1)

G. Campbell, R. K. Kostuk, J. Opt. Soc. Am A 12, 1113 (1995).
[CrossRef]

J. Opt. Soc. Am. B (1)

Proc. SPIE (1)

J. P. Eblen, W. J. Gunning, D. Taber, P. Yeh, M. Khoshnevisan, J. Beedy, L. Hale, Proc. SPIE 2262, 234 (1994).
[CrossRef]

SID Digest (1)

J. P. Eblen, W. J. Gunning, J. Beedy, D. Taber, L. Hale, P. Yeh, M. Khoshnevisan, SID Digest 94, 245 (1994).

Other (6)

M. Bass ed., Handbook of Optics, 2nd ed. (McGraw-Hill, New York, 1994), Vol. 2, pp. 33.66–33.69.

S.-T. Wu, “Nematic liquid crystals for active optics,” in Optical Materials— A Series of Advances, S. Musikant, B. J. Thompson, eds. (Dekker, New York, 1990), Vol. 1; S.-T. Wu, C.-S. Wu, W. Warenghem, M. Ismaili, Proc. SPIE 1815, 179 (1992).
[CrossRef]

C. Gu, P. Yeh, in Digest of Annual Meeting (Optical Society of America, Washington, D.C., 1995), paper ThHH2.

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988); A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984); and references therein.

L. M. Brekhovskikh, Waves in Layered Media (Academic, New York, 1960); M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980); and references therein.

P. Yeh, W. J. Gunning, J. P. Eblen, M. Khoshnevisan, “Compensator for liquid crystal displays having two types of layers with different refractive indices alternating,” U.S. patent5,196,953 (1994).

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Figures (2)

Fig. 1
Fig. 1

Normal surfaces of the TM and the TE waves, where K and β are normalized by ω/c. Solid curves correspond to the exact Bloch wave solutions [Eqs. (2)]. Dashed–dotted curves in (a) correspond to the lowest-order approximation [Eqs. (3) and (4)]. Dotted curves in (b) correspond to the higher-order approximation [approximations (7) and (8)].

Fig. 2
Fig. 2

Dispersion of form birefringence (a) without and (b) with the intrinsic material dispersion.

Equations (10)

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E = E K ( z ) exp [ i ( ω t - β x - K z ) ] ,
cos ( K Λ ) = cos ( k 1 z a ) cos ( k 2 z b ) - 1 2 ( k 2 z k 1 z + k 1 z k 2 z ) × sin ( k 1 z a ) sin ( k 2 z b ) ( TE ) , cos ( K Λ ) = cos ( k 1 z a ) cos ( k 2 z b ) - 1 2 ( n 2 2 k 1 z n 1 2 k 2 z + n 1 2 k 2 z n 2 2 k 1 z ) × sin ( k 1 z a ) sin ( k 2 z b ) ( TM ) ,
K 2 n o 2 + β 2 n o 2 = ( ω c ) 2 ( TE ) , K 2 n o 2 + β 2 n e 2 = ( ω c ) 2 ( TM ) ,
n o 2 = a Λ n 1 2 + b Λ n 2 2 ,             1 n e 2 = a Λ 1 n 1 2 + b Λ 1 n 2 2 .
K 2 + β 2 ( n o ω c ) 2 + ( a b ) 2 ( n 1 2 - n 2 2 ) 2 ( ω / c ) 4 12 Λ 2 ( TE ) ,
K 2 + β 2 { n o 2 n e 2 + ( a b ) 2 ( n 1 2 - n 2 2 ) 2 ( 1 n 1 2 + 1 n 2 2 ) [ 2 ( ω / c ) 2 - ( 1 n 1 2 + 1 n 2 2 ) β 2 ] 12 Λ 2 } ( n o ω c ) 2 + ( a b ) 2 ( n 1 2 - n 2 2 ) 2 ( ω / c ) 4 12 Λ 2 ( TM ) .
n ˜ o 2 n o 2 + ( Δ n o 2 ) ,
n ˜ e 2 n e 2 + ( Δ n o 2 ) n e 4 ( n 1 2 + n 2 2 ) 10 n 1 2 n 2 2 n o 2 n e 2 + ( Δ n o 2 ) 5 ,
( Δ n o 2 ) = ( a b ) 2 ( n 1 2 - n 2 2 ) 2 ( ω / c ) 2 12 Λ 2 ;
Δ n ˜ ( Λ 0 ) Δ n ( Λ 0 ) = ( n ˜ o - n ˜ e ) Λ 0 ( n o - n e ) Λ 0 1 + 4 π 2 n 2 a b 15 λ 2 ,

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